Linus Torvalds used Minix as a 'digital textbook' for writing the Linux kernel (as did many other developers), no one is denying that. In that sense, Linux is based on Minix. Minix forms the base of Linux because Linus studied Minix, and then went on to create Linux.

However, to say that Linux 'thanks its existence' to Minix is an overstatement. Linux and Minix are fundamentally different-- Where Minix is a microkernel, Linux is a monolithic kernel. Indeed, the exact opposite.